präsentation alexander hasse

Synthesis of Compliant Mechanisms via topology optimization

Alexander Hasse

Summer School ASME 2016

Outline

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Introduction What is topology optimization? A short introduction to structural optimization Topology Optimization of compliant

Mechanisms Design examples

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Introduction

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Prof. Dr. sc. ETH Alexander HasseFriedrich-Alexander-UniversityD-91052 ErlangenGermany

Phone: 0049 9131 85 23663Email: hasse@mfk.fau.de

Alexander Hasse

2001-2007 Diploma Degree in Mechanical Engineering (TU Dresden)

2007-2011 Doctoral Work in MechanicalEngineering (ETH Zurich)

2011-2012 Post-Doc (ETH Zurich)

2012-2014 Head Engineering (Monolitix AG)

Since 2014 Professor for Mechatronic Systems at FAU Erlangen-Nuermberg

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IntroductionRobotic grippers

4Source: Monolitix

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IntroductionMedical instruments

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surgical handle – conventional

surgical handle – compliant

Source: Empa

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IntroductionShape adaptive structures

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What is topology optimization?The design problem in general

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What is topology optimization?Definition of topology, shape and size

8Source: Wikipedia, Bendsoe and Sigmund

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What is topology optimization?Example force inverter

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inF outF

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Structural optimizationDifferent modules

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Structural modelling

Opti. formulation

Req

uire

men

ts

Optimization algorithm

Parameterization

min ( )

( ) 0

dfmitV K

,

x

φx

Source: Kress

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Structural optimizationParameterization and structural modelling

11Source: Bendsoe and Sigmund

0( )

1

mi

ii

xk k

Ground-structure approach

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Structural optimizationParameterization and structural modelling

12Source: Bendsoe and Sigmund

0( )

1

mp ii

i

xk k

Solid Isotropic Material with Penalisation (SIMP)

0( )

1

mi

ii

m x v

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Optimization formulation„Spring method“ according to Bendsoe and Sigmund

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Aktor

in inF u

outu outu

ink outk

inF

0( )

1

min

min ( )

0 1 , 1,...,

out

mi

ii

i

u

mit

x v V

x x i m

xx

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Preprocessing

1. Build up ground structure

2. Generate „ground“ stiffness matrix for each ground-structure member

„Spring method“Example force inverter

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symmetry conditionfixed stiffness ground-element fixed stiffness ground-element

0( ) , 1i i mk

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„Spring method“Example force inverter

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Optimization

Initial design variables

Built stiffness matrix

Caculate structural response

Update design variables by a proper optimization algorithm

0( )

1

mi

i in outi

xk k k k

0x

1inu k f

outu

newx

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„Spring method“Example force inverter

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Update design variables by a proper optimization algorithmoutu

Sensitivity analysis

Calculate sensitivities

newx

out

i

dudx

( ) ( )out out i out i

i

du u x x u xdx x

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„Spring method“Example force inverter

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Calculate sensitivities analytically

ku f

d d ddx dx dxu f kk u

1T Toutdu d d ddx dx dx dx

u f kz z k u

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Structural optimizationProcedure

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Structural modelling

Opti. formulation

Req

uire

men

ts

Optimization algorithm

Parameterization

min ( )

( ) 0

dfmitV K

,

x

φx

Source: Kress

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Structural optimization

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Recommendation

inF outF

“A 99 line topology optimization code written in Matlab” written by Sigmundwith small modifications described in the book „Topology Optimization -Theory, Methods, and Applications”

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Optimization formulation„MPE/SE“ according to Ananthasuresh

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inF

outu

inF

(a) (b) (c)virtuellF

TMPE v ku

virtuellkv f

inku f

TSE u ku

inku fmin MPEf

SE

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Optimization formulation„Characteristic stiffness formulation“ according to Wang

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inF

outu(a)

0aa ac a a

ca cc c

é ù é ù é ùê ú ê ú ê ú=ê ú ê ú ê úë ûë û ë û

k k u fk k u

1( )aa ac cc ca a a a-- = =k k k k u ku f

11 12

21 22

in in

out out

F uk kF uk k

2( )11 22( ) e dGA GAf k kx

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Formulation for mechanisms with multiple outputs

NACA 0012

NACA 2412

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6Design criterion

kφ λφ

1Tφ φ

1 1 12 2 2

T Tφ kφ λφ φ λ

u φ

2 12

TSE φ kφ

2

12

T SEφ kφ

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6Design criterion

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6Design criterion

1 0 00 0

0 01 1 10 02 2 2

0 00 0

T

p

l

l

é ùê úê úê úê úê ú= = ê úê úê úê úê úê úë û

Φ kΦ K

1k

2 3, , , nk

1 1φ χ

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6Design criterion

1 , 1

1 , 1

Ti iTi i

i m

i m p

= " =

= " = +

χ χ

χ χ

1

1

0 00

1 1 12 2 2

00 0

mT

m

p

k

kk

k

+

é ùê úê úê úê úê úC C = = ê úê úê úê úê úê úë û

k K

1 1m m p+é ùC = ë ûχ χ χ χ

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6Optimization formulation

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6Optimization formulation

aa ac a a

ca cc c c

k k u fk k u f

1( )aa ac cc cak k k k k

kφ λφ

1 dφ φ

Problem

mitkψ wψ 1 dψ φ

Solution

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6Optimization formulation

1 1 1ˆ | | ... | | | ... |d j j qΦ φ φ φ φ φ

with1...

maxdj dii qaa

1 dψ φ

1 22 2 1

1 1

T

T

ψ kφψ φ ψψ kψ

1

1

Tqi q

q q iTi i i

ψ kφψ φ ψ

ψ kψ

1 2| | ... | qΨ ψ ψ ψ

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6Optimization formulation

1 0 00 0

0 00 0

T

q

Λ Ψ kΨ

withkψ wψ 1 dψ φ

1

2..

( )min q

f x

Objective function

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6Design example – force inverterProblem statement and parameterization

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symmetry conditionfixed stiffness ground-element fixed stiffness ground-element

Design domain with support and master DoFs; arrows define the directions of positive displacement in the master DoFs

Parameterized design domain

1

( )n

i ii

xk x k

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6Design example – force inverterResults and discussion

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6Design example – force inverterEigenvalue analysis of the force inverters

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6Design example – force inverterDeformation plots

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6Design Example – shape adaptive structureProblem statement

35

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6Design Example – shape adaptive structureResults

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Load case 1, F =-140 N Load case 2, Fx =140 N, Fy =-140 N

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6Design example – Morphing wing

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d

=k Eigenvalue problem

= ( )Belt Stiffk k k xStiffness matrix

Design criterion

1 d

1 2

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6Design example – Morphing wing

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NACA 0012

NACA 2412

d

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6Design example – Morphing wing

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S

IP

LE

T

xx

xxx

= ( )Belt Stiffk k k x

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6Design example – Morphing wingOptimization procedure

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6Design Example – Morphing wing

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Output from the design routine

Prototype as hybrid design: CFRP belt and inner stiffening structure in Polyamide

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6Design example – Morphing wing

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Single Belt Complete Belt‐Rib Structure

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6Experimental investigations

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Applied displacement at the trailing edge

Applied displacement at the leading edge

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6Experimental investigations

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Comparison between the desired and the actual deformed shapes by an applied displacement of 3.5 mm

Comparison between the desired and the actual deformed shapes by an applied displacement of 3.5 mm

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Thank you for your attention.